DP State for Counting

Define dp[v] = number of ways to reach node $v$ from the source. Base case: dp[source] = 1 (one way to stay at source), all others start at $0$. Process nodes in topological order. For each edge $(u, v)$, add dp[u] to dp[v]. Why addition?

Each path to $u$ can extend to $v$ by following edge $(u, v)$. So the total number of paths to $v$ is the sum of paths from all nodes that have edges to $v$.